The word "rain" is an activity noun. Most activity nouns do not take "the" as an article. Rain, (along with a few other activity nouns) is an exception and takes "the." When we say "the rain" we are talking about the weather activity. When we say "rain" we mean the actual water. (There is a song "Singing in The rain." Watch it on youtube, it's a classic. http://youtu.be/w40ushYAaYA)
Here are some examples of activity nouns that don;t take "the" in sentences: "I like going to church." "I was sent to jail." "I'm tired. I think I'll go to bed." Here are some of the exceptions: "I need to see the doctor." "If you want answers, you should go to the library." "Can you find avocados at the store during Winter?"
So, your sentence works best with "the," and I would write it thus: "Alicia was standing under the tree during the rain." However, I think it works better if you use the simple past, "Alicia stood under the tree during the rain."
"Whilst" is used chiefly in BrE. It introduces a "background activity" in which to place an event. "While I was writing this answer, my dog begged for attention." "While" cannot be placed before "the rain" in your sentence because it is not a preposition but a subordinate conjunction (http://englishplus.com/grammar/00000377.htm). If you want to use "while" (and you can) this is a possibility: "While it rained, Alicia was standing under the tree." I prefer, "While it rained, Alicia stood under the tree."
Hope this helps. Ask in the comments section if you need some follow-up.